Alternating Current (AC) periodically reverses direction, unlike direct current (DC). AC is the standard form of electricity supplied to homes and industries because it can be efficiently stepped up or down using transformers.
Key Concepts
- Representation: An AC voltage is v = Vm sin(omega t), where Vm is the peak (maximum) voltage, omega = 2 pi f is the angular frequency, and f is the frequency (Hz).
- RMS Values: The Root Mean Square (RMS) value is the effective DC equivalent. Vrms = Vm / √(2) ≈ 0.707 Vm. Similarly Irms = Im / √(2).
- Resistor in AC circuit: Voltage and current are in phase. I = Vm / R × sin(omega t). Power P = Vrms × Irms = Irms2 R.
- Inductor in AC circuit: Current lags voltage by pi/2 (90°). Inductive reactance XL = omega L = 2 pi f L. Irms = Vrms / XL.
- Capacitor in AC circuit: Current leads voltage by pi/2 (90°). Capacitive reactance XC = 1 / (omega C). Irms = Vrms / XC.
- Phase relation mnemonic: "CIVIL" — In a Capacitor (C), I leads V; in an Inductor (L), V leads I.
- Series RLC Circuit: Z = √(R2 + (XL - XC)2). Phase angle phi: tan(phi) = (XL - XC)/R. Irms = Vrms / Z.
- Resonance: When XL = XC, impedance Z is minimum (= R), current is maximum. Resonant frequency: omega0 = 1/√(LC); f0 = 1 / (2 pi √(LC)).
- Power in AC circuit: Average power P = Vrms Irms cos(phi), where cos(phi) is the power factor. For pure L or C: phi = 90°, so P = 0. For R: phi = 0°, P = Vrms Irms.
- Wattless current: The component of current that does not contribute to power is called wattless current (imaginary part); it exists in L and C circuits.
- Transformer: Works on mutual induction. Vs/Vp = Ns/Np = Ip/Is. Step-up: Ns > Np; step-down: Ns < Np. For ideal transformer, Pprimary = Psecondary.
- Efficiency of transformer: eta = Pout/Pin × 100%. Losses include copper loss (I2R in windings) and core loss (hysteresis + eddy currents).
- Key Formulas
- Vrms = Vm/√(2); Irms = Im/√(2)
- XL = omega L; XC = 1/(omega C)
- Z = √(R2 + (XL - XC)2)
- Resonance: f0 = 1/(2 pi √(LC))
- Power: P = Vrms Irms cos(phi)
- Transformer: Vs/Vp = Ns/Np
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The peak voltage of an AC source is 220 √(2) V. Find the RMS voltage and frequency given omega = 100 pi rad/s.
Vrms = Vm / √(2) = 220 √(2) / √(2) = 220 V
f = omega / (2 pi) = 100 pi / (2 pi) = 50 Hz
A 10 Omega resistor is connected to 100 V RMS AC supply. Find the RMS current and average power.
Irms = Vrms / R = 100/10 = 10 A
P = Irms2 × R = 100 × 10 = 1000 W
Find the inductive reactance of a 0.1 H inductor at 50 Hz.
XL = 2 pi f L = 2 pi × 50 × 0.1 = 10 pi ≈ 31.4 Omega
Find the capacitive reactance of a 100 µF capacitor at 50 Hz.
XC = 1/(2 pi f C) = 1/(2 pi × 50 × 100 × 10-6) = 1/(pi × 10-2) = 100/pi ≈ 31.8 Omega
A series RLC circuit has R = 5 Omega, L = 0.1 H, C = 100 µF connected to 100 V RMS at 50 Hz. Find Z, Irms, and power.
XL = 31.4 Omega; XC = 31.8 Omega
Z = √(52 + (31.4 - 31.8)2) = √(25 + 0.16) ≈ 5.016 Omega
Irms = 100/5.016 ≈ 19.9 A
cos(phi) = R/Z = 5/5.016 ≈ 0.997
P = Vrms × Irms × cos(phi) = 100 × 19.9 × 0.997 ≈ 1985 W
Find the resonant frequency for L = 4 mH and C = 10 µF.
f0 = 1/(2 pi √(LC)) = 1/(2 pi √(4 × 10-3 × 10 × 10-6)) = 1/(2 pi √(4 × 10-8)) = 1/(2 pi × 2 × 10-4) = 1/(4 pi × 10-4) ≈ 796 Hz
A step-up transformer converts 220 V to 11000 V. The primary has 200 turns and primary current is 5 A. Find secondary turns and secondary current (assuming ideal).
Ns/Np = Vs/Vp = 11000/220 = 50; Ns = 50 × 200 = 10000 turns
Is = Ip × Np/Ns = 5 × 200/10000 = 0.1 A
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Common mistakes
- Using peak values instead of RMS values when calculating power — always use Vrms and Irms for average power.
- Forgetting that XL increases with frequency while XC decreases with frequency.
- At resonance, impedance Z = R (not zero); current is maximum but finite.
Summary
AC circuits use phasors to represent the phase relationships between voltage and current. Resistors, inductors, and capacitors have different phase behaviours. Impedance governs current in RLC circuits, with maximum current at resonance. Transformers use electromagnetic induction to step voltages up or down efficiently.